Mathematical Research Letters 2, (1995) MODULARITY OF A FAMILY OF ELLIPTIC CURVES. Fred Diamond and Kenneth Kramer

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1 Mathematical Research Letters 2, (1995) MODULARITY OF A FAMILY OF ELLIPTIC CURVES Fred Diamond and Kenneth Kramer We shall explain how the following is a corollary of results of Wiles [W]: Theorem. Suppose that E is an elliptic curve over Q all of whose 2- division points are rational, i.e., an elliptic curve defined by y 2 =(x a)(x b)(x c) for some distinct rational numbers a, b and c. Then E is modular. Recall that Wiles proves that if E is a semistable elliptic curve over Q, then E is modular [W,Thm. 0.4]. He begins by considering the Galois representations ρ E, 3 (respectively, ρ E, 3 ) on the 3-division points (respectively,3-adic Tate module) of E. If ρ E, 3 is irreducible,then a theorem of Langlands and Tunnell is used to show that ρ E, 3 arises from a modular form. Wiles deduces that ρ E, 3 also arises from a modular form by appealing to his results in [W,Ch. 3] and those with Taylor in [TW] to identify certain universal deformation rings as Hecke algebras. This suffices to prove that E is modular if ρ E, 3 is irreducible. When ρ E, 3 is reducible, Wiles gives an argument which allows one to use ρ E, 5 instead. In fact,wiles results apply to a larger class of elliptic curves than those which are semistable [W,Thm. 0.3],and this was subsequently extended in [Di] to include all elliptic curves with semistable reduction at 3 and 5. Rubin and Silverberg noted that an elliptic curve as in the above theorem necessarily has a twist which is semistable outside 2,and hence,is modular by [Di,Thm. 1.2]. The purpose of this note is to explain how,by a refinement of their observation,the above theorem follows directly from Wiles work,without appealing to [Di]. Lemma 1 (Rubin-Silverberg). By at most a quadratic twist, an elliptic curve as in the theorem may be brought to the form (1) E : y 2 = x(x A)(x + B) for some nonzero integers A and B with A and B relatively prime, B even and A 1mod4. Let C = A + B. For odd primes p, the curve E has Received March 24,

2 300 FRED DIAMOND AND KENNETH KRAMER good reduction at p if p is prime to ABC and multiplicative reduction at p otherwise. Proof. Note that a curve as in the theorem is isomorphic to one defined by equation (1) for some integers A and B with AB(A + B) 0. Let D = gcd(a, B). Twisting by Q( D),we may assume that A and B are relatively prime. By translating x or exchanging A and B,we may assume that B is even. Finally,if A 1 mod 4,we twist again by Q(i). The reduction type of E for odd primes p may be determined as in [Se2, 4] and [Si1,Ch. VII]. See [O, I.1] for discussion of the reduction type and conductor of curves given by equation (1),but under certain restrictions in the case p = 2. See also [Da,Lemma 2.1] for a related case. We treat the reduction type at p = 2 in the following lemma. Lemma 2. Suppose that E is an elliptic curve over Q 2 defined by the model ( 1), with A 1mod4and B even. The reduction type, conductor exponent f 2 (E) and valuation of the minimal discriminant of E are given by the following table: ord 2 (B) ν 5 Kodaira Symbol III I1 III I 0 I 2ν 8 f 2 (E) ord 2 ( min ) ν 8 Proof. A twist of E by the unramified extension Q 2 ( A) affects neither reduction type nor conductor exponent,and provides a model of the form (2) y 2 = x(x + 1)(x + s) with ord 2 (s) = ord 2 (B) 1 and discriminant = 16s 2 (1 s) 2. For the convenience of the reader,we indicate the appropriate translations of model,depending on ord 2 (s),so that the explicit criteria of Tate s algorithm [T] may be used. If ord 2 (s) = 1,then ord 2 ( ) = 6. Put y + x for y in (2) to get (3) y 2 +2xy = x 3 + sx 2 + sx. If ord 2 (s) = 2,then ord 2 ( ) = 8. Put x + 2 for x in (3),to get y 2 +2xy +4y = x 3 +(s +6)x 2 +(5s + 12)x +(6s +8). If ord 2 (s) = 3,use the model (3) with ord 2 ( ) = 10. If ord 2 (s) 4,the model (3) is not minimal and may be reduced to (4) y 2 + xy = x 3 + s 4 x2 + s 16 x

3 MODULARITY OF A FAMILY OF ELLIPTIC CURVES 301 with discriminant s 2 (1 s) 2 /256. Thus,(4) has good reduction if ord 2 (s) = 4 and multiplicative reduction if ord 2 (s) 5. To show that an elliptic curve over Q is modular,we may replace it with one to which it is isomorphic over Q. We may therefore assume that E is defined by equation (1) with A and B as in Lemma 1. If E has good or multiplicative reduction at p = 2,then E is semistable and we can conclude from [W,Thm. 0.4] that E is modular. In view of Lemma 2,we may therefore also assume,henceforth,that ord 2 (B) =1, 2or3. Let l be an odd prime. Choose a basis for E[l],the kernel of multiplication by l on E,and let ρ E, l denote the representation G Q GL 2 (F l ) defined by the action of G Q = Gal ( Q/Q) one[l]. For each prime p,we fix an embedding Q Q p and regard G p = Gal ( Q p /Q p ) as a decomposition subgroup of G Q at a place over p. Thus, ρ E, l G p is equivalent to the representation of G p defined by its action on E[l]( Q p ). Let I p G p denote the inertia group. Recall the special role played by the prime l = 3 in Wiles approach. We simply write ρ for ρ E, 3. If ρ is irreducible,then ρ is modular by the theorem of Langlands and Tunnell (see [W,Ch. 5]). Since E has good or multiplicative reduction at 3,we need only verify certain hypotheses on ρ in order to apply [W,Thm. 0.3] to conclude that E is modular. We shall see that if E has additive reduction at p = 2,then those hypotheses are satisfied,the crucial point being the verification of a local condition at p = 2. The irreducibility of ρ in this case is a byproduct of our verification. In fact,we have the following stronger result: Lemma 3. If ord 2 (B) =1, 2 or 3 and l is an odd prime, then ρ E, l I 2 is absolutely irreducible. Proof. For the moment,consider the more general case of a representation ψ : I SL 2 ( F l ),where I is the inertia group of a finite Galois extension of p-adic fields and l p is a prime. Let b(ψ) denote the wild conductor exponent [Se2, 4.9]. If b(ψ) is odd,then ψ is irreducible. Indeed,were ψ to be reducible,it would be equivalent to a representation of the form ( ) χ 0 χ 1. But then,because b is integer-valued and additive on short exact sequences, b(ψ) = 2b(χ) would be even.

4 302 FRED DIAMOND AND KENNETH KRAMER Under the hypotheses of this lemma,the elliptic curve E has additive reduction at 2 and odd conductor exponent f 2 (E) =2+b( ρ E, l I 2 ),independent of the choice of odd prime l. Since det ρ E, l G 2 is an unramified character associated to Q 2 (µ l ),the image of I 2 under ρ E, l is contained in SL 2 (F l ). It follows that ρ E, l I 2 is absolutely irreducible. Remark. When Lemma 3 applies,an analysis of the group structure of SL 2 (F 3 ) shows that the image of wild ramification at p = 2 under ρ,and hence, ρ E, l,for any odd l,is isomorphic to the quaternion group of order 8. Under the hypotheses of Lemma 3,we see that even the restriction of ρ = ρ E, 3 to Gal ( Q/Q(µ 3 )) is absolutely irreducible. Using Lemma 3,one can also easily check the local conditions on ρ appearing as hypotheses in [W,Thm. 0.3]. Since it is left to the reader of [W] to verify that those local conditions are sufficient to apply the central result [W,Thm. 3.3], we shall explain directly how this is done in the case with which we are concerned. Again,we consider,more generally, ρ E, l for odd primes l. First recall that ρ E, l is unramified at p if p l is a prime of good reduction,i.e.,if p does not divide labc. Next we recall how the Tate parametrization is used to describe ρ E, l G p for primes p at which E has multiplicative reduction (see [Se2, 2.9]). Let F denote the unramified quadratic extension of Q p in Q p. Then E has split multiplicative reduction over F and the Tate parametrization (see [Si2, V.3]) provides an isomorphism Q p /q Z = E( Qp ) of Gal ( Q p /F )-modules for some q Q p with ord p (q) > 0. From this it follows that for each prime l,there is a filtration of Gal ( Q p /F )-modules 0 Z l (1) T l (E) Z l 0, where T l (E) is the l-adic Tate module and Z l (1) = lim µ l n( Q p ). One then checks that the representation of G p on T l (E) is equivalent to one of the form ( ) ɛ χ 0 1 where χ is either trivial or the unramified quadratic character of G p and ɛ is the cyclotomic character given by the action of G p on Z l (1). It follows that the representation of G p on E[l] is of this form as well,but with ɛ now defined by the action of G p on µ l. Suppose now that p l is an odd prime dividing ABC. Then the above analysis of multiplicative reduction applies to ρ E, l G p and shows that ρ E, l is either unramified or type (A) at p in the terminology of [W,

5 MODULARITY OF A FAMILY OF ELLIPTIC CURVES 303 Ch. 1]. (The first possibility occurs precisely when ord p (ABC) is divisible by l; see [Se2, 4].) Suppose next that p = l. If p divides ABC,then the above analysis of multiplicative reduction shows that ρ E, l G p is ordinary at p in the terminology of [W,Ch. 1]. If on the other hand p does not divide ABC, then the elliptic curve E has good reduction at p. In fact,the equation (1) defines an elliptic curve E over Z p such that E Qp is isomorphic to E Qp (see [Si2, IV.5]). The kernel of multiplication by l on E is a finite flat group scheme E[l] over Z p. The representation ρ E, l G p is given by the action of G p on E[l]( Q p ),which we may identify with E[l]( Q p ). In this sense, ρ E, l G p arises from a finite flat group scheme over Z p. Now ρ E, l G p is reducible if and only if E has ordinary reduction at p,i.e.,if and only if E Fp is ordinary. In that case ρ E, l is ordinary at p in the sense of [W]. On the other hand, ρ E, l G p is irreducible if and only if E Fp is supersingular, in which case ρ E, l is flat at p in the sense of [W,Ch. 1]. Finally,suppose that p = 2 and E has additive reduction at 2. Then ord 2 (B) =1, 2or3,and ρ E, l I 2 is absolutely irreducible by Lemma 3. We claim that ρ E, l G 2 is of type (C) at 2 in the terminology of Wiles [W, Ch. 1]. Recall that this means that H 1 (G 2,W) = 0,where W is the G 2 - module of endomorphisms of E[l]( Q 2 ) of trace zero. From the triviality of the local Euler characteristic ([Se1,Thm. II.5]),we have #H 1 (G 2,W)=#H 0 (G 2,W) #H 2 (G 2,W). By local Tate duality ([Se1,Thm. II.1]),we have #H 2 (G 2,W)=#H 0 (G 2,W ) where W = Hom(W, µ l ). Therefore,we wish to prove that H 0 (G 2,W) and H 0 (G 2,W ) both vanish. But in fact H 0 (I 2,W) and H 0 (I 2,W ) already vanish. Indeed, I 2 acts trivially on µ l,from which we deduce that there is a (noncanonical) isomorphism W = W of I2 -modules; hence,it suffices to show that H 0 (I 2,W) = 0. Since I 2 acts absolutely irreducibly on F 2 l,schur s lemma implies that the only I 2-invariant endomorphisms of F 2 l are scalars. But the only scalar in W is zero. Specializing to the case l = 3,we now conclude that the representation ρ E, 3 of G Q on T 3 (E) arises from a modular form. Indeed,Wiles [W,Thm. 3.3] establishes an isomorphism between the universal deformation ring of type D and the Hecke algebra T D,where D =(, Σ, Z 3, ) with as flat or Selmer according to whether or not E has supersingular reduction at 3; Σ as the set of primes dividing 3ABC.

6 304 FRED DIAMOND AND KENNETH KRAMER Since ρ E, 3 defines a deformation of ρ of type D,the universal property of the deformation ring thus provides a homomorphism T D Z 3 with the following property: for all p not dividing 3ABC,the Hecke operator T p is sent to a p = p +1 N p where N p is the number of F p -points on the reduction of E mod p. The definition of T D ensures that this homomorphism arises from a normalized eigenform of weight two whose p th Fourier coefficient is a p for all such p. Hence E is modular. Acknowledgements The authors are grateful to Kevin Buzzard,Ken Ribet and Karl Rubin for comments on an earlier version of this note. References [Da] H. Darmon, The equations x n + y n = z 2 and x n + y n = z 3, Duke IMRN 10 (1993), [Di] F. Diamond, On deformation rings and Hecke rings, preprint. [O] J. Oesterlé, Nouvelles approches du théorème de Fermat, Séminaire Bourbaki 694 ( ), Astérisque (1988), [Se1] J. P. Serre, Cohomologie Galoisienne, 5 e éd, LNM 5. Springer-Verlag, New York, [Se2], Sur les représentations modulaires de degré 2 de Gal ( Q/Q), Duke Math. J. 54 (1987), [Si1] J. Silverman, The arithmetic of elliptic curves, GTM 106, Springer-Verlag, New York, [Si2], Advanced topics in the arithmetic of elliptic curves, GTM 151, Springer- Verlag, New York, [T] J. T. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, in Modular Functions of One Variable IV, LNM 476, Springer-Verlag, New York, [TW] R. Taylor and A. Wiles, Ring theoretic properties of certain Hecke algebras, to appear in Ann. Math. [W] A. Wiles, Modular elliptic curves and Fermat s Last Theorem, to appear in Ann. Math. D.P.M.M.S., 16 Mill Lane, Univ. of Cambridge, Cambridge, CB2 1SB, UK address: f.diamond@pmms.cam.ac.uk Department of Mathematics, Queens College (CUNY), Flushing, NY address: kramer@qcvaxa.acc.qc.edu